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Shades of Fermat's Last Theorem

The familiar Pythagorean 3-4-5 triple gives one solution to (x-1)^n + x^n = (x+1)^n so what about other solutions for x an integer and n= 2, 3, 4 or 5?

Upsetting Pitagoras

Find the smallest integer solution to the equation 1/x^2 + 1/y^2 = 1/z^2

BT.. Eat Your Heart Out

If the last four digits of my phone number are placed in front of the remaining three you get one more than twice my number! What is it?

Lattice Points

Age 16 to 18 Challenge Level:

(1) Show that if there is one lattice point (point with integer coordinates) on the parabola $$y=ax^2$$ then there are infinitely many.

(2) Find all the lattice points on the hyperbola $$x^2 - y^2 = 84.$$